Entanglement, geometry and the Ryu Takayanagi formula

Juan Maldacena

Kyoto, 2013

Aitor Lewkowycz

Tom Faulkner

Lewkowycz, JM ArXiv:1304.4926 & Faulkner, Lewkowycz, JM, to appear

Previously argued by Fursaev

Entanglement entropy

( Not a proper observable in the sense of Dirac, not a linear function of the density matrix)

Entanglement for subregions in QFTs

BB A B

It is divergent, but divergencies are well understood. Divergencies are universal. We will often talk about finite quantities. Usually difficult to compute. Encodes interesting dynamical information c, f, a theorems

Bombelli, Koul, Lee, Sorkin Srednicki…..

Entanglement in theories with gravity duals

A

Minima area surface in the bulk

Ryu-Takayanagi

Leading order in GN expansion

Half space

Half of space

A B

Euclidean time

By a Weyl transformation on the metric we can map this to

The entaglement entropy is equal to the thermodynamic entropy

Circle:

Casini, Huerta, Myers

Replica trick

Usual thermodynamic computation

Bulk dual = Black brane with a hyperbolic horizon

Callan Wilczek

More general regions

BB

B

A

We still have a tau circle. Choose a Weyl factor so that the circle is not contractible in the boundary metric. The metric is tau dependent.

Bulk problem

Finding a sequence of bulk geometries where the boundary has the previous geometry but with

AdS3 Faulkner (from CFT Hartman)

Typically the circle shrinks smoothly in the interior

Generalized gravitational entropy

U(1) invariant Not U(1) invariant Gibbons Hawking

n=5 Z5 symmetry in the bulk

Evaluating the gravitational action we can compute the entropy using the replica trick

The Ryu Takayanagi conjecture reduces to proving the following classical statement: The above combination of derivatives of the analytic continuation in n of the gravitational actions gives the area of the minimal surface. This is a minimal codimension two surface such that the tau circle shrinks at its location We will argue for this statement by giving the analytic continuation of the geometries.

The analytic continuation of the geometries

Analytic continuation = Same for all values of n For integer n non singular in covering space. Here no clear covering space… We just evaluate the action, with no contribution from the singularity and multiply by n

With no contribution from the singularity !!

Equation for the surface

As n 1 we have the spacetime geometry produced by a cosmic string. Equations of motion from expanding Einstein’s equations near the singularity and demanding that the geometry of the singularity is not modified.

Equations for a minimal surface

Unruh, Hayward, Israel, Mcmanus Boisseau, Charmousis, Linet

Evaluating the action

We only want to evaluate it for n close to one.

- =

- + -

n=1

n=1

Zero by equations of motion of the n=1 solution Manifestly depends only on the tip

n≠1

n≠1 n≠1flattend n≠1flattend

Hawking, Fursaev, Solodukhin

-

Curvature of the flattened solution near the tip

This is multiplied by the area of the rest of the dimensions Got the usual area formula

n≠1 n≠1flattend

Comments

• This works if there is a moment of time reflection symmetry. (e.g. spatial regions in static spacetimes).

• We did not prove the more general HRT formula, valid for dynamical situations.

• Higher curvature action ? Should work, details ?

Hubeny, Rangamani Takayanagi

Hung, Myers, Smolkin Fursaev, Patrushev, Solodukhin Chen, Zhang, Bhattacharyya, Kaviraj, Sinha

Quantum corrections

• So far we have discussed the terms of order 1/GN

• What about the first quantum correction in the bulk.

• We will motivate it with a puzzle.

Mutual information

A B Large separation

But

Wolf, Verstraete, Hastings, Cirac

Headrick Takayanagi

• No problem

• I(A,B)=0 only to leading order in 1/N or GN .

• Then it should be non-zero at order N0 or GN0

Direct computation

• Compute all one loop determinants around all the smooth integer n solutions.

• Continue in n 1

• Is there a simpler formula for the final answer?

AdS3 : Barrella, Dong,. Hartnoll, Martin Faulkner

Quantum correction

A

AB

Define a bulk region AB , inside the minimal surface. Compute the entanglement of the bulk quantum fields between AB and the rest of the spacetime.

The …

Correction to the area due to the one loop change in the bulk metric. Area in the quantum corrected metric.

Quantum expectation value of the formal expression of the area.

The counterterms that render the one loop bulk quantum theory finite can lead to additional terms. They are of the usual Wald form.

Some interesting cases

KS theory

Theory with a mass gap for most degrees of freedom but with some remaining massless fields

A

AB

Bulk computation effectively four dimensional after KK reduction only massless part contributes non-trivially at long distances.

Mutual information

A B

AB BB

Leading long distance entanglement = entanglement of the spins.

Mutual Information ``OPE’’

General story in a CFT

A B

Pairs of operators (this comes from operators on different replicas) Analytic continuation of the OPE

coefficients for the replicas.

Headrick Calabrese, Cardy Tonni

(not a true OPE in the original space )

In the holographic case

A B Large separation AB BB

On the boundary we expect the previous OPE In the bulk we also have an entanglement computation with two well separated regions. Bulk theory is not conformal and it is in curved space. However, we still expect some OPE where we exchange pairs of bulk particles. (For conformally coupled bulk fields we can do it more precisely) By GKPW bulk particles two point functions of operators. Same structure for the OPE !

Thermal situations

AB . . . .

. .

. . . .

Thermal gas in AdS contribution from the entropy of the gas

Derivation

Here ρ is the bulk density matrix, after propagation in tau for 2 π

Conclusions

• We proposed a simple formula for the quantum corrections of RT

• We checked it in a few cases where the quantum corrections are the dominant effect

• Further checks... ? More interesting checks …?

General comments

• There is an interesting connection between black hole entropy and entanglement entropy.

• Precise formulation of the Bekenstein formula

• Proof of the generalized 2nd law. Wall

Casini arXiv:0804.2182 Inspired by Marolf, Minic, Ross